Analysis and implementation of new fractional-order multi-scroll hidden attractors

To improve the complexity of chaotic signals,in this paper we first put forward a new three-dimensional quadratic fractional-order multi-scroll hidden chaotic system,then we use the Adomian decomposition algorithm to solve the proposed fractional-order chaotic system and obtain the chaotic phase diagrams of different orders,as well as the Lyaponov exponent spectrum,bifurcation diagram,and SE complexity of the 0.99-order system.In the process of analyzing the system,we find that the system possesses the dynamic behaviors of hidden attractors and hidden bifurcations.Next,we also propose a method of using the Lyapunov exponents to describe the basins of attraction of the chaotic system in the matlab environment for the first time,and obtain the basins of attraction under different order conditions.Finally,we construct an analog circuit system of the fractional-order chaotic system by using an equivalent circuit module of the fractional-order integral operators,thus realizing the 0.9-order multi-scroll hidden chaotic attractors.     

分数阶共轭Chen混沌系统中的混沌及其电路实验仿真

研究了分数阶共轭Chen混沌系统的混沌行为, 给出了分数阶共轭Chen系统存在混沌吸引子的必要条件. 并设计了一种传递函数的新电路单元用来实现分数阶共轭Chen混沌系统, 利用Multisim2001电子工作平台进行了电路实验仿真. 数值模拟和电路实验仿真结果相符, 证实了分数阶共轭Chen系统确实存在混沌现象和所设计电路单元的有效性. 关键词： 分数阶 共轭Chen混沌系统 电路单元 电路仿真     

分数阶Lorenz系统的分析及电路实现

频域传递函数近似方法不仅是常用的 分数阶混沌系统相轨迹的数值分析方法之一, 而且也是设计分数阶混沌系统电路的主要方法. 应用该方法首先研究了分数阶Lorenz系统的混沌特性, 通过对Lyapunov指数图、分岔图和数值仿真分析, 发现了其较为丰富的动态特性, 即当分数阶次从0.7到0.9以步长0.1变化时, 该分数阶Lorenz系统既存在混沌特性, 又存在周期特性, 从数值分析上说明了在更低维的Lorenz系统中存在着混沌现象. 然后又基于该方法和整数阶混沌电路的设计方法, 设计了一个模拟电路实现了该分数阶Lorenz系统, 电路中的电阻和电容等数值是由系统参数和频域传递函数近似确定的. 通过示波器观测到了该分数阶Lorenz系统的混沌吸引子和周期吸引子的相轨迹图, 这些电路实验结果与数值仿真分析是一致的, 进一步从物理实现上说明了其混沌特性. 关键词： 分数阶系统 Lorenz系统 分岔分析 电路实现     

一种具有隐藏吸引子的分数阶混沌系统的动力学分析及有限时间同步

对于具有隐藏吸引子的混沌系统,既有文献大多只针对整数阶系统进行分析与控制研究.基于Sprott E系统,构建了仅有一个稳定平衡点的分数阶混沌系统,通过相位图、Poincare映射和功率谱等,分析了该系统的基本动力学特征.结果显示,该系统展现出了丰富而复杂的动力学特性,且通过随阶次变化的分岔图可知,系统在不同阶次下呈现出周期运动、倍周期运动和混沌运动等状态,这些动力学特征对于保密通信等实际工程领域有重要的研究价值.针对该具有隐藏吸引子的分数阶系统,应用分数阶系统有限时间稳定性理论设计控制器,对系统进行有限时间同步控制,并通过数值仿真验证了其有效性.     

One to four-wing chaotic attractors coined from a novel 3D fractional-order chaotic system with complex dynamics

A novel 3D fractional-order chaotic system is proposed in this paper. And the system equations consist of nine terms including four nonlinearities. It's interesting to see that this new fractional-order chaotic system can generate one-wing, two-wing, three-wing and four-wing attractors by merely varying a single parameter. Moreover, various coexisting attractors with respect to same system parameters and different initial values and the phenomenon of transient chaos are observed in this new system. The complex dynamical properties of the presented fractional-order systems are investigated by means of theoretical analysis and numerical simulations including phase portraits, equilibrium stability, bifurcation diagram and Lyapunov exponents, chaos diagram, and so on. Furthermore, the corresponding implementation circuit is designed. The Multisim simulations and the hardware experimental results are well in accordance with numerical simulations of the same system on the Matlab platform, which verifies the correctness and feasibility of this new fractional-order chaotic system.     

Realization of fractional-order Liu chaotic system by a new circuit unit

A new circuit unit for the analysis and the synthesis of the chaotic behaviours in a fractional-order Liu system is proposed in this paper. Based on the approximation theory of fractional-order operator, an electronic circuit is designed to describe the dynamic behaviours of the fractional-order Liu system with α = 0.9. The results between simulation and experiment are in good agreement with each other, thereby proving that the chaos exists indeed in the fractional-order Liu system.     

A new piecewise linear Chen system of fractional-order:Numerical approximation of stable attractors

In this paper we present a new version of Chen's system:a piecewise linear(PWL) Chen system of fractional-order.Via a sigmoid-like function,the discontinuous system is transformed into a continuous system.By numerical simulations,we reveal chaotic behaviors and also multistability,i.e.,the existence of small parameter windows where,for some fixed bifurcation parameter and depending on initial conditions,coexistence of stable attractors and chaotic attractors is possible.Moreover,we show that by using an algorithm to switch the bifurcation parameter,the stable attractors can be numerically approximated.     

Realization of fractional-order Liu chaotic system by circuit

In this paper, chaotic behaviours in the fractional-order Liu system are studied. Based on the approximation theory of fractional-order operator, circuits are designed to simulate the fractional- order Liu system with $q=0.1-0.9$ in a step of 0.1, and an experiment has demonstrated the 2.7-order Liu system. The simulation results prove that the chaos exists indeed in the fractional-order Liu system with an order as low as 0.3. The experimental results prove that the fractional-order chaotic system can be realized by using hardware devices, which lays the foundation for its practical applications.     

Transient transition behaviors of fractional-order simplest chaotic circuit with bi-stable locally-active memristor and its ARM-based implementation

This paper proposes a fractional-order simplest chaotic system using a bi-stable locally-active memristor. The characteristics of the memristor and transient transition behaviors of the proposed system are analyzed, and this circuit is implemented digitally using ARM-based MCU. Firstly, the mathematical model of the memristor is designed, which is nonvolatile, locally-active and bi-stable. Secondly, the asymptotical stability of the fractional-order memristive chaotic system is investigated and some sufficient conditions of the stability are obtained. Thirdly, complex dynamics of the novel system are analyzed using phase diagram, Lyapunov exponential spectrum, bifurcation diagram, basin of attractor, and coexisting bifurcation, coexisting attractors are observed. All of these results indicate that this simple system contains the abundant dynamic characteristics. Moreover, transient transition behaviors of the system are analyzed, and it is found that the behaviors of transient chaotic and transient period transition alternately occur. Finally, the hardware implementation of the fractional-order bi-stable locally-active memristive chaotic system using ARM-based STM32F750 is carried out to verify the numerical simulation results.     

Three-dimensional chaotic autonomous van der pol–duffing type oscillator and its fractional-order form

In this paper, a three-dimensional autonomous Van der Pol-Duffing (VdPD) type oscillator is proposed. The three-dimensional autonomous VdPD oscillator is obtained by replacing the external periodic drive source of two-dimensional chaotic nonautonomous VdPD type oscillator by a direct positive feedback loop. By analyzing the stability of the equilibrium points, the existence of Hopf bifurcation is established. The dynamical properties of proposed three-dimensional autonomous VdPD type oscillator is investigated showing that for a suitable choice of the parameters, it can exhibit periodic behaviors, chaotic behaviors and coexistence between periodic and chaotic attractors. Moreover, the physical existence of the chaotic behavior and coexisting attractors found in three-dimensional proposed autonomous VdPD type oscillator is verified by using Orcard-PSpice software. A good qualitative agreement is shown between the numerical simulations and Orcard-PSpice results. In addition, fractional-order chaotic three-dimensional proposed autonomous VdPD type oscillator is studied. The lowest order of the commensurate form of this oscillator to exhibit chaotic behavior is found to be 2.979. The stability analysis of the controlled fractional-order-form of proposed three-dimensional autonomous VdPD type oscillator at its equilibria is undertaken using Routh–Hurwitz conditions for fractional-order systems. Finally, an example of observer-based synchronization applied to unidirectional coupled identical proposed chaotic fractional-order oscillator is illustrated. It is shown that synchronization can be achieved for appropriate coupling strength.     

Dynamical analysis of fractional-order Rössler and modified Lorenz systems

This Letter is devoted to the dynamical analysis of fractional-order systems, namely the Rössler and a modified Lorenz system. The work here described compares the dynamical regimes of such fractional-order systems to that of the corresponding standard systems. It turns out that most of the chaotic attractors are topologically equivalent to those found in the original integer-order systems, although in some particular (and apparently rare) cases unusual bifurcation patterns and attractors are found.     

Aspects of improved heat conduction relation and chemical processes in 3D Carreau fluid flow

A new fourth-order memristor chaotic oscillator is taken to investigate its fractional-order discrete synchronisation. The fractional-order differential model memristor system is transformed to its discrete model and the dynamic properties of the fractional-order discrete system are investigated. A new method for synchronising commensurate and incommensurate fractional discrete chaotic maps are proposed and validated. Numerical results are established to support the proposed methodologies. This method of synchronisation can be applied for any fractional discrete maps. Finally the fractional-order memristor system is implemented in FPGA to show that the chaotic system is hardware realisable.     

A specific state variable for a class of 3D continuous fractional-order chaotic systems

A specific state variable in a class of 3D continuous fractional-order chaotic systems is presented.All state variables of fractional-order chaotic systems of this class can be obtained via a specific state variable and its (q-order and 2q-order) time derivatives.This idea is demonstrated by using several well-known fractional-order chaotic systems.Finally,a synchronization scheme is investigated for this fractional-order chaotic system via a specific state variable and its (q-order and 2q-order) time derivatives.Some examples are used to illustrate the effectiveness of the proposed synchronization method.     

Generating Multidirectional Variable Hidden Attractors via Newly Commensurate and Incommensurate Non-Equilibrium Fractional-Order Chaotic Systems

This article investigates a non-equilibrium chaotic system in view of commensurate and incommensurate fractional orders and with only one signum function. By varying some values of the fractional-order derivative together with some parameter values of the proposed system, different dynamical behaviors of the system are explored and discussed via several numerical simulations. This system displays complex hidden dynamics such as inversion property, chaotic bursting oscillation, multistabilty, and coexisting attractors. Besides, by means of adapting certain controlled constants, it is shown that this system possesses a three-variable offset boosting system. In conformity with the performed simulations, it also turns out that the resultant hidden attractors can be distributively ordered in a grid of three dimensions, a lattice of two dimensions, a line of one dimension, and even arbitrariness in the phase space. Through considering the Caputo fractional-order operator in all performed simulations, phase portraits in two- and three-dimensional projections, Lyapunov exponents, and the bifurcation diagrams are numerically reported in this work as beneficial exit results.     

Adaptive control and synchronization of a fractional-order chaotic system

In this paper, the chaotic dynamics of a three-dimensional fractional-order chaotic system is investigated. The lowest order for exhibiting chaos in the fractional-order system is obtained. Adaptive schemes are proposed for control and synchronization of the fractional-order chaotic system based on the stability theory of fractional-order dynamic systems. The presented schemes, which contain only a single-state variable, are simple and flexible. Numerical simulations are used to demonstrate the feasibility of the presented methods.     

Chaos in fractional-order generalized Lorenz system and its synchronization circuit simulation

The chaotic behaviours of a fractional-order generalized Lorenz system and its synchronization are studied in this paper. A new electronic circuit unit to realize fractional-order operator is proposed. According to the circuit unit, an electronic circuit is designed to realize a 3.8-order generalized Lorenz chaotic system. Furthermore, synchronization between two fractional-order systems is achieved by utilizing a single-variable feedback method. Circuit experiment simulation results verify the effectiveness of the proposed scheme.     

基于Lyapunov方程的分数阶新混沌系统的控制

新混沌系统是一种不同于Lorenz混沌系统、Chen混沌系统以及Liu混沌系统的新的三阶连续自治混沌系统.本文基于波特图的频域近似方法,提出了一种混合型电路单元来近似实现分数阶算子,并设计电路实现了2·7阶新混沌系统.基于Lyapunov方程的系统稳定性判定理论,设计了相应的控制器,实现了对分数阶新混沌系统的控制.     

分数阶Liu混沌系统及其电路实验的研究与控制

基于波特图的频域近似方法,研究了分数阶Liu混沌系统,并设计了一种树形电路单元来实现分数阶Liu混沌系统,通过对2.7阶Liu混沌系统的电路仿真和实验,以及α=0.8—0.1(步长0.1)Liu混沌系统的电路仿真,验证了树形电路单元的有效性,证实分数阶Liu混沌系统中确实存在混沌现象,且存在混沌的最低阶数为0.3. 设计简单有效的线性反馈控制器,实现了分数阶Liu混沌系统的混沌控制. 关键词： 分数阶Liu系统 电路实验 混沌控制     

基于Julia分形的多涡卷忆阻混沌系统

忆阻器作为一种非线性电子元件,能用作混沌系统中的非线性项,从而提高系统的复杂度.分形与混沌是密切相连的,分别对两者的研究都已成熟,却鲜有将分形过程应用到混沌系统中,以产生丰富的混沌吸引子.为了探索将分形与混沌系统相结合的可能性,本文首先提出了一个新的忆阻混沌系统,并从对称性、耗散性、平衡点稳定性、功率谱、Lyapunov指数和分数维等方面探讨了系统的动力学特性;紧接着,把经典的Julia分形过程应用到该忆阻混沌系统中,产生了新的混沌吸引子,并将几种由Julia分形衍生的变形Julia分形过程应用于文中提出的忆阻混沌系统,获得了丰富的混沌吸引子;最后,讨论了分形过程中的复常数对系统的影响.从仿真结果可以看出,分形过程与混沌系统的结合能产生丰富的多涡卷混沌吸引子.这不仅为产生多涡卷混沌吸引子提供了一种新方法,还弥补了使用功能函数方法造成混沌系统不光滑的不足.     

Designing synchronization schemes for fractional-order chaotic system via a single state fractional-order controller

In this paper the synchronization of fractional-order chaotic systems is studied and a new single state fractional-order chaotic controller for chaos synchronization is presented based on the Lyapunov stability theory. The proposed synchronized method can apply to an arbitrary three-dimensional fractional chaotic system whether the system is incommensurate or commensurate. This approach is universal, simple and theoretically rigorous. Numerical simulations of several fractional-order chaotic systems demonstrate the universality and the effectiveness of the proposed method.